An efficient preconditioner for evolutionary partial differential equations with $\theta$-method in time discretization
Abstract
In this study, the -method is used for discretizing a class of evolutionary partial differential equations. Then, we transform the resultant all-at-once linear system and introduce a novel one-sided preconditioner, which can be fast implemented in a parallel-in-time way. By introducing an auxiliary two-sided preconditioned system, we provide theoretical insights into the relationship between the residuals of the generalized minimal residual (GMRES) method when applied to both one-sided and two-sided preconditioned systems. Moreover, we show that the condition number of the two-sided preconditioned matrix is uniformly bounded by a constant that is independent of the matrix size, which in turn implies that the convergence behavior of the GMRES method for the one-sided preconditioned system is guaranteed. Numerical experiments confirm the efficiency and robustness of the proposed preconditioning approach.
Cite
@article{arxiv.2408.03535,
title = {An efficient preconditioner for evolutionary partial differential equations with $\theta$-method in time discretization},
author = {Yuan-Yuan Huang and Po Yin Fung and Sean Y. Hon and Xue-Lei Lin},
journal= {arXiv preprint arXiv:2408.03535},
year = {2024}
}